Multijoints and Factorisation

We solve the dual multijoint problem and prove the existence of so-called "factorisations" for arbitrary fields and multijoints of \(k_j\)-planes. More generally, we deduce a discrete analogue of a theorem due in essence to Bourgain and Guth. Our result is a universal statement which describes a property of the discrete wedge product without any explicit reference to multijoints and is stated as follows: Suppose that \(k_1 + \ldots + k_d = n\). There is a constant \(C=C(n)\) so that for any field \(\mathbb{F}\) and for any finitely supported function \(S : \mathbb{F}^n \rightarrow \mathbb{R}_{\geq 0}\), there are factorising functions \(s_{k_j} : \mathbb{F}^n\times \mathrm{Gr}(k_j, \mathbb{F}^n)\rightarrow \mathbb{R}_{\geq 0}\) such that $$(V_1 \wedge\cdots\wedge V_d)S(p)^d \leq C\prod_{j=1}^d s_{k_j}(p, V_j),$$ for every \(p\in \mathbb{F}^n\) and every tuple of planes \(V_j\in \mathrm{Gr}(k_j, \mathbb{F}^n)\), and $$\sum_{p\in \pi_j} s(p, e(\pi_j)) =||S||_d$$ for every \(k_j\)-plane \(\pi_j\subset \mathbb{F}^n\), where \(e(\pi_j)\in \mathrm{Gr}(k_j,\mathbb{F}^n)\) denotes the translate of \(\pi_j\) that contains the origin and \(\wedge\) denotes the discrete wedge product.